Unlike the previous analytical expression of the quantum Fisher information matrix which required the diagonalization of the density matrix, Šafránek [1]} has recently provided another analytical expression using the density matrix in the Liouville space formalism. Technically, by employing the vectorization of the density matrix \(\rho \) , the Lyapunov equations (REF ) are converted to a linear system that can be solved without diagonalizing the matrix \(\rho \) . This approach has proven to be successful and can be used for any finite dimensional density matrix. This is based on using the vec-operator which allows us to transform a density matrix into a column vector. Indeed, the operator \({\rm vec}\left[A\right]\)  for any matrix \(A\in \emph {M}^{n\times n}\)  in the Liouville space, where \(\emph {M}^{n\times n}\)  refers to the space of \(n\times n\)  martrices, is given by
\({\rm vec}\left[A \right]=\left(a_{11},...,a_{n1},a_{12},...,a_{n2},a_{1n},...,a_{nn} \right)^{T}.\) 
